At any point in the triangle, the distances to A, B, and C must add up to 1.0. Along the line connecting A and C, B must be zero. At any apex, one component is 1.0 while both others are zero. Note the numbers on the sides leaving the apex for C. The fraction of C must decrease both along the line toward A and the line toward B. The following applet lets you select a point and read the concentrations of A, B, and C. Please be prepared to do this by hand on a quiz.
The following is in terms understandable by those who have studied physical chemistry. In the context of solvent extraction, the possibilities are one phase in which the solvents are miscible or two liquid phases. For just one phase, F=3+2-1, or four degrees of freedom. One degree goes to fix the temperature or pressure, another for taking a slice from the three-dimensional diagram, and the remaining two degrees mean that one phase takes an area of the two-dimensional plot. When two liquid phases exist, F=3+2-2. Now there is one remaining degree of freedom; this is a line on the two-dimensional plot extending from the composition of one of the immiscible liquids to the composition of the other immiscible liquid. These lines define an envelope in which any point is a composition that must split into the two compositions of the two liquid phases. There will be a point where the line constricts to a point known as the plait point.